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Grid generation

In this section we discuss ways of generating a suitable computational grid. As we stated previously in section 1.9 this subject is highly a matter of experience but since we are modeling the flow as a parabolic flow, ie a flow in which the flow state at one z-level is independent of the states at subsequent levels we can, for instance, calculate the grid in a step by step manner while we calculate the solution. In this fashion we actually construct a solution adaptive grid which could be used to keep the truncation error below some user specified level. For completeness we give an example of a solution adaptive grid generator algorithm

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Estimate a new suitable steplength possibly by utilizing the estimated local truncation error a the current point.

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Calculate the solution at the new point.

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Estimate certain derivatives of the dependent variable(s), calculate the local truncation error at the new point and compare to the user specified accuracy (resolution).

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If the resolution requirement is not met calculate a new smaller steplength by taking the values of the derivatives and the order of the method into consideration--return to step 2. If the requirement is fulfilled precede with the next point--return to step 1.

In order to keep the code developing at a minimum we will not put such an effort into the grid generation. Instead we calculate the grid in advance, ie before we attempt to calculate the solution. We will make the following plausible requirements to a suitable computational grid:

• Steplengths which do not succeed a grid spacer by a distance of less than $\Delta z_{\mbox{\protect\scriptsize sp}}$ (see section 6.7.2.3) is determined by considering the value of the linear heat generation rate, q'. The details of the steplength calculation are illuminated later.
• A number of steplengths which immediately succeed a spacer is distributed uniformly over the range of the spacer pressure variation width, $\Delta z_{\mbox{\protect\scriptsize sp}}$.

We will discuss these two requirements in turn in the following two subsections.

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